Determining reaction order and k from a graph

The rate law (equation) shows how rate depends on the concentration of reacting species A and gives important information about the pathway for the reaction.  

Zero order 
reaction rate = k[A]0 = k

First order
reaction rate = k[A]

Second order
reaction rate = k[A]2

The exponent on [A] (reaction order) and the rate constant, k, must be determined experimentally using reliable methods. 
This involves making several measurements at different concentrations. 

Both the first and second order rate equations have a linear form (y = mx) where a change in variable y produces a corresponding change in variable x. This suggests that both the order and the rate constant can be determined using several experiments and plotting
 
reaction rate on the y-axis
[A]order on the x-axis (order = 1 or 2)
  
For reactions where rate depends on [A], a plot of 
rate against [A] is linear
only if the reaction is first order. 
rate against [A]2 is linear
only if the reaction is second order.

The slope of the line is then equal to the rate constant k.